Physics – Problem Set
(ii) Use the relation La = (r ⇥ P)a = ✏abcrbPc (where r = (x, y, z) and P = i~(@x, @y, @z))
together with the identity
✏abc✏ade = bdce becd (8) (most easily demonstrated by enumerating the possibilities) to show that
L2 =@rr2@r r2r2 (9)
and that as a result, the operator L2/r2 is the angular part of the Laplacian in spherical polar coordinates. Note that the La only contain derivatives in the angular directions, because the rotation generators L = ir ⇥ r only contain derivatives in the directions orthogonal to the radial direction.
3. Finding the eigenfunctions of the angular Laplacian
(i) From the previous problem, the angular parts of waves in 3d can be written in a
of eigenbfausnisctions of L2 and Lz. However, since Lx and Ly don’t commute with Lz, they map eigenfunctions
L2fμm(✓,)=μfμm(✓,) , Lzfμm(✓,)=mfμm(✓,) (10)
to other eigenfunctions with in general di↵erent eigenvalues of Lz – in other words, thought of as a giant matrix, these operators are not diagonal in the basis of fμm but rather have o↵-diagonal matrix elements that send you to a di↵erent eigenfunction. The eigenvalue of L2 will remain the same, however, since Lx and Ly commute with L2. In fact, use the commutation relations of the rotation generators to show that
Lz⇣(Lx ± iLy)fμm⌘ = (m ± 1)⇣(Lx ± iLy)fμm⌘ (11) L2⇣(Lx ± iLy)fμm⌘ = μ⇣(Lx ± iLy)fμm⌘ (12)
where one consistently takes either the upper sign or the lower sign throughout. Thus the operators L± = Lx ± iLy move us around in the space of eigenvalues of Lz keeping the eigenvalue μ of L2 fixed. Show that the angular Laplacian can be written in the forms
L2 =LL+ +L2z +Lz =L+L +L2z Lz (13) (ii) Recalling that Lz = i@ and that w ⌘ x+iy = rei, we see that w` = r`ei` are
eigenfunctions of Lz with eigenvalue m = `. Show that
L+ w` = 0 (14)
and that as a consequence these functions are also eigenfunctions of L2 with the eigenvalue μ = `(` + 1). Similarly show that w ̄` = r`ei` is an eigenfunction of L2 and Lz with eigenvalues μ = `(` + 1) and m = `, respectively; and that it is annihilated by L.
4. Spherical Bessel functions and radial waves
(i) The radial part of the 3d wave equation in spherical polar coordinates is
⇣1@rr2@r +`(`+1)+k2⌘f`k(r)=0 r2 r2
where k2 = !2/vs2. Show that the substitution
f`k(r) = pkrF`+1/2(kr)
1 turns the radial wave equation into Bessel’s equation.
(ii) Verify by direct substitution that the two solutions for ` = 0 are f`k(r) = f`(kr) = exp[±ikr]
up to an overall normalization constant. Thus we conclude that (up to normalization) (1) exp[+iu] (2) exp[iu]
are the corresponding pair of Bessel functions (often called Hankel functions just to confuse you). Away from the origin, the two complex exponentials represent radially incoming and outgoing waves, since when we combine with the time dependence ei!t the radial waves are exp[ik(vst ± r)]/r. If the domain U 2 R where we are solving the equation includes the origin, we should select the nonsingular solution sin[kr]/r which has the wave bounce o↵ the origin with Neumann boundary conditions.
(iii) Show that the Bessel equation can be factored in two di↵erent ways
H1/2(u) = pu , H1/2(u) = pu
⇣@u + ⌫ + 1 ⌘⇣@u ⌫ ⌘ F⌫ (u) = F⌫ (u) ⇣ u⌘⇣ u⌘
@u ⌫1 @u +⌫ F⌫(u)=F⌫(u) (21) uu
and that as a result ⇣@u u⌫ ⌘F⌫ = c⌫+1F⌫+1 (22)
where c⌫+1 is some constant. Hint: Apply the operator on the LHS to the Bessel equation and rearrange terms.
Use this relation to find H (1) (u) and H (1) (u), up to overall normalization. 3/2 5/2
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